We study the role of connectivity of communication networks in privatecomputations under information theoretic settings. It will be shown thatsome functions can be computed by private protocols even if theunderlying network is 1-connected but not 2-connected. Then we give acomplete characterisation of non-degenerate functions that can becomputed on non-2-connected networks.

Furthermore, a general technique for simulating private protocols onarbitrary networks will be presented. Using this technique every privateprotocol can be simulated on arbitrary $k$-connected networks using onlya small number of additional random bits.

Finally, we give matching lower and upper bounds for the number ofrandom bits needed to compute the parity function on $k$-connectednetworks.